Optimal. Leaf size=305 \[ -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (d+e x)^{9/2} (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-9 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)}+\frac {3 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-9 c d g+c e f)}{4 e^2 \sqrt {d+e x} (2 c d-b e)}-\frac {3 c (4 b e g-9 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2 \sqrt {2 c d-b e}} \]
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Rubi [A] time = 0.48, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {792, 662, 664, 660, 208} \begin {gather*} -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (d+e x)^{9/2} (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-9 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)}+\frac {3 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-9 c d g+c e f)}{4 e^2 \sqrt {d+e x} (2 c d-b e)}-\frac {3 c (4 b e g-9 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2 \sqrt {2 c d-b e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 662
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(c e f-9 c d g+4 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{4 e (2 c d-b e)}\\ &=\frac {(c e f-9 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(3 c (c e f-9 c d g+4 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{8 e (2 c d-b e)}\\ &=\frac {3 c (c e f-9 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) \sqrt {d+e x}}+\frac {(c e f-9 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(3 c (c e f-9 c d g+4 b e g)) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=\frac {3 c (c e f-9 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) \sqrt {d+e x}}+\frac {(c e f-9 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {1}{4} (3 c (c e f-9 c d g+4 b e g)) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {3 c (c e f-9 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) \sqrt {d+e x}}+\frac {(c e f-9 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {3 c (c e f-9 c d g+4 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 \sqrt {2 c d-b e}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 129, normalized size = 0.42 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac {c (d+e x)^2 (4 b e g-9 c d g+c e f) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{e (b e-2 c d)^2}+\frac {5 d g}{e}-5 f\right )}{10 e (d+e x)^{9/2} (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.45, size = 228, normalized size = 0.75 \begin {gather*} \frac {\sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (4 b e g (d+e x)-2 b d e g+2 b e^2 f+4 c d^2 g+5 c e f (d+e x)-4 c d e f-13 c d g (d+e x)-8 c g (d+e x)^2\right )}{4 e^2 (d+e x)^{5/2}}-\frac {3 \left (-4 b c e g+9 c^2 d g+c^2 (-e) f\right ) \tan ^{-1}\left (\frac {\sqrt {b e-2 c d} \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2}}{\sqrt {d+e x} (b e+c (d+e x)-2 c d)}\right )}{4 e^2 \sqrt {b e-2 c d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 998, normalized size = 3.27 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{3} e f + {\left (c^{2} e^{4} f - {\left (9 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} f - {\left (9 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} - {\left (9 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (c^{2} d^{2} e^{2} f - {\left (9 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (8 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g x^{2} - {\left (2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - 2 \, b^{2} e^{3}\right )} f + {\left (34 \, c^{2} d^{3} - 21 \, b c d^{2} e + 2 \, b^{2} d e^{2}\right )} g - {\left (5 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (58 \, c^{2} d^{2} e - 37 \, b c d e^{2} + 4 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (2 \, c d^{4} e^{2} - b d^{3} e^{3} + {\left (2 \, c d e^{5} - b e^{6}\right )} x^{3} + 3 \, {\left (2 \, c d^{2} e^{4} - b d e^{5}\right )} x^{2} + 3 \, {\left (2 \, c d^{3} e^{3} - b d^{2} e^{4}\right )} x\right )}}, -\frac {3 \, {\left (c^{2} d^{3} e f + {\left (c^{2} e^{4} f - {\left (9 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} f - {\left (9 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} - {\left (9 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (c^{2} d^{2} e^{2} f - {\left (9 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) + \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (8 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g x^{2} - {\left (2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - 2 \, b^{2} e^{3}\right )} f + {\left (34 \, c^{2} d^{3} - 21 \, b c d^{2} e + 2 \, b^{2} d e^{2}\right )} g - {\left (5 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (58 \, c^{2} d^{2} e - 37 \, b c d e^{2} + 4 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (2 \, c d^{4} e^{2} - b d^{3} e^{3} + {\left (2 \, c d e^{5} - b e^{6}\right )} x^{3} + 3 \, {\left (2 \, c d^{2} e^{4} - b d e^{5}\right )} x^{2} + 3 \, {\left (2 \, c d^{3} e^{3} - b d^{2} e^{4}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 665, normalized size = 2.18 \begin {gather*} \frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (12 b c \,e^{3} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-27 c^{2} d \,e^{2} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+3 c^{2} e^{3} f \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+24 b c d \,e^{2} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-54 c^{2} d^{2} e g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+6 c^{2} d \,e^{2} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+12 b c \,d^{2} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-27 c^{2} d^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+3 c^{2} d^{2} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-8 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,e^{2} g \,x^{2}+4 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,e^{2} g x -29 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c d e g x +5 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,e^{2} f x +2 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b d e g +2 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,e^{2} f -17 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,d^{2} g +\sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c d e f \right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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